The NP-hard Colorful Components problem is, given a vertex-colored graph, to delete a minimum number of edges such that no connected component contains two vertices of the same color. It has applications in multiple sequence alignment and in multiple network alignment where the colors correspond to species. We initiate a systematic complexity-theoretic study of Colorful Components by presenting NP-hardness as well as fixed-parameter tractability results for different variants of Colorful Components. We also perform experiments with our algorithms and additionally develop an efficient and very accurate heuristic algorithm clearly outperforming a previous min-cut-based heuristic on multiple sequence alignment data.